How can a DQN backpropagate its loss?
It is quite common in DQN to instead of having the neural network represent function $f(s,a) = \hat{q}(s,a,\theta)$ directly, it actually represents $f(s)= [\hat{q}(s,1,\theta), \hat{q}(s,2,\theta), \hat{q}(s,3,\theta) . . . \hat{q}(s,N_a,\theta)]$ where $N_a$ is the maximum action, and the input the current state. That is what is going on here. It is usually done for a performance gain, since calculating all values at once is faster than individually.

However, in a Q learning update, you cannot adjust this vector of output values for  actions that you did not take. You can do one of two things:


Figure out the gradient due to the one item with a TD error, and propagate that backwards. This involves inserting a known gradient into the normal training update step in a specific place and working from there. This works best if you are implementing your own backpropagation with low-level tools, otherwise it can be a bit fiddly figuring out how to do it in a framework like Keras.
Force the gradients of all other items to be zero by setting the target outputs to be whatever the learning network is currently generating.


If you are using something like Keras, the second approach is the way to go. A concrete example where you have two networks n_learn and n_target that output arrays of Q values might be like this:


For each sample (s, a, r, next_s, done) in your minibatch*


Calculate array of action values from your learning network qvals = n_learn.predict(s)
Calulate TD target for $(s,a)$ e.g. td_target = r + max(n_target.predict(next_s)) (discount factor and how to handle terminal states not shown)
Alter the one array item that you know about from this sample qvals[a] = td_target
Append s to your train_X data and qvals to your train_Y data

Fit the minibatch n_learn.fit(train_X, train_Y)




* It is possible to vectorise these calculations for efficiency. I show it as a for loop as it is simpler to describe that way